AD-A240 898

Report No. NADC-91077-60

H0 OPTIMAL CONTROL THEORY OVER

A FINITE HORIZON

M. Bala SubrahmanvamAir Vehicle and Crew Systems Techmology DepartmentNAVAL AIR DEVELOPMENT CENTERWarminster, PA 18974-5000

SEPTEMBER 1991 TI

INTERIM REPORT UPeriod Covering January 1991 to August 1991Task No. R36240000AWork Unit No. 101598Program Element No. 0602936N

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I0602936N 0 1R36240000A1 10159811 TITLE (include Security Classification)

Hc~ Optimal Control Theory over a Finite Horizon

12 PERSONAL AUTriORiS;

M. Bala Subrahmanyam13a TYPE OF REPORT 3cTINPE CC)jEPED 4 DA-- O PEFOP- Year Month Day) 5 PAC: O.

Interim cov 1/91 To 8/91 1991 September 2616 SUPPLEMENTARY NOTATION,

7 (0517l- CODES 18 SAAB)CT TEPVS \ContinueC an reverse if necessary and odrntf, b) block number)

FIELD GPO .jP SUB GPO. H,, Optimal Control, Infinal H,,, Norm, Finite04 Horizon, Existence Theorem, Suboptimal Control

19 ABSTR ACT (Continue on reverse if necessary and identify by block number)

In this report a finite horizon H.. optimal control problem is treated. Results guaran-teeing the existence of a unique optimal control and the worst exogenous input are derived.A criterion for the evaluation of the infimnal H,,, norm is then given in terms of the leastpositive value for which a certain boundary value problem has a nontrivial solution. Oncethe infimnal value is known, a noninfimal value can be selected and suboptimal H', con-trollers can be synthesized. The problem of synthesizing suboptimal H,_ controllers is alsoconsidered in a very general case. Without making use of any transformations, expressionsfor the output feedback controller are derived in terms of solutions of two dynamic Riccatiequations. In the time-invariant case, the solutions of these equations usually converge toconstant matrices.

20 DISTP BuTIOrN AV(A;AB J'Y OfAC'C 21 ARYPAC T SE CUP (4 C0A~C' J.

[RUNCLASSIc ED'JLIV -ED El~f AS P 7j - r , , Unclassified22a NA^IV'E O ' ArSPOS8, %DF N. LA 2 ri n (uo AreC I~e 12,r c- (.

M. Bala Subrahmanyan (215) 441-7151 Code 60123D Form 1473, JUN 86 Pre,,ous editions are cbsolete (~(P. :7K-

S/N (I10O2-LT-01-6603 Unclassified

NADC-91077-60

CONTENTS

1. INTRODUCTION ............ ........................... 1

2. EXISTENCE OF OPTIMAL INPUTS ....... .................... 2

3. INFIMAL Ho, NORM ........... ......................... 7

4. COMPUTATION uF A......... ......................... 9

5. A DIFFERENTIAL EQUATION FOR A ...... .................. .10

6. EXAMPLES .............. ............................. 12

7. SUBOPTIMAL PROBLEM FORMULATION ....................... .14

8. FULL STATE FEEDBACK PROBLEM ....... ................... .. 15

9. OUTPUT FEEDBACK CONTROLLER ......... ................... 18

10. SUMMARY OF RESULTS ......... ....................... .. 23

11. CONCLUSIONS ............. ........................... 25

12. FUTURE WORK ............ ........................... 25

REFERENCES .......................................... .26

Accession For

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NADC-91077-60

1. INTRODUCTION

There is a current need for a robust multivariable flight control design techniquewhich gives good performance under changing environment, disturbances, and uncertaintyin modeling. Also, it is very useful to know the maximum possible performance underworst case conditions. Design techniques based on Hc methods are ideally suited foryielding good performance of the aircraft even under worst case conditions. Thus, at theNaval Air Development Center, efforts are underway to demonstrate the feasibility andadvantages of flight control design techniques based on finite horizon H, techniques.

The H,, optimal control problem has received considerable attention recently and itis well-known that H,, suboptimal controllers can be synthesized via the solution of twoalgebraic Riccati equations under certain restrictive assumptions[l]. These assumptionscan be removed using various transformations[2], and thus controller synthesis can beaccomplished in the general case in an indirect manner. Recently the techniques havebeen extended to time-varying systems[3] under similar restrictive assumptions. One ofthe contributions of the present report is the derivation of the results in a general case inthe time-varying setting. The synthesis of the controller is accomplished for a general errorcriterion in the finite horizon case and the implementation of the equations on a digitalcomputer is easy. The synthesis can be accomplished by means of two dynamic Riccatiequations, one related to the controller part and the other to the observer part. In thetime-invariant case, the solutions of these equations usually tend to constant matrices.

A lot of current research has also been directed towards the problem of estimatingthe infimal H, norm of a given system. There are a few iterative techniques in the time-invariant case. There are virtually no techniques in the time-varying case and anothercontribution of this report is an efficient technique for the estimation of the infimal H,norm in a very general setting. The technique consists of considering the inherent minimaxproblem and treating the adjoint variables associated with the maximization problem asstate variables for the minimization problem. Once this is accomplished, the techniques of[4-9] can be applied to get the infimal norm.

We treat the problem of existence and computation of the minimal H" norm initially.Then the problem of synthesizing the suboptimal H, controllers will be taken up. In thederivation of the output feedback controller, the full state feedback expressions are usefulin the definition of the gain of the controller part and duality plays an important role inthe assignment of the gain of the observer.

In all control problems it is very useful to know the maximum possible performanceunder worst case conditions. The theory of Sections 2-6 is useful in obtaining a quantitativeidea of achievable performance. Since suboptimal design is more practical, this topic willbe treated in Sections 7-10. Also the command following problem can be suitably recastto fit the problem formulation of Section 7.

In the time-invariant case, the solutions of the dynamic Riccati equations involvedusually tend towards constant matrices, if the final time is large enough. This is very

1

NADC-91077-60

valuable in the case of flight control design. Research is also being done to successfully

deal with parameter variations under the present setting. In our reserach we are mainlyinterested in the robust performance aspects of the aircraft under variations of the systemmodel, as opposed to robust stability considerations.

2. EXISTENCE OF OPTIMAL INPUTSIn this section we obtain results for the existence of optimal exogenous and control

inputs for the finite horizon Ho, problem. Consider the system given by

i = A(t)x + Bi(t)u + B 2 (t)v. x(O) = 0. ,

z = C(t)x + D(t)u + E(t)v, (2.2)

where x and z are the state vector and the error vector respectively. The matricesA(t), B1 (t), B 2 (t), C(t), D(t), and E(t) will be assumed to be continuous on [0, TI, where

T is the final time. In additica u,v E L2 (0, T).The problem is to show the existence of u and v for which

To v*/, dtinf sup f (2.3)to u f T z*l dt

is achieved. In (2.3) R(t) and W(i) are continuous positive definite matrices on [0, T].

We now consider the maximization part in (2.3). We will show that given any v,

there exists a u which minimizes foT *Wz dr. By simple changes in variables, the aboveminimization problem can be converted to a problem of the form considered in Example2, Section 3.3 of [10]. From the results of [10], there exists a unique u which minimizesfT z*Wz dt.

We can write the functional in (2.3) as

J(u, V) =foT v*(t)R(t)v(t) dt (2.4)fo. x.W, + x*lV 2 u + low u + *Vv + l v + 01V6 0}

We will use the adjoint variables associated with the maximization part of (2.3) as statevariables for the minimization part of (2.3).

LetA = inf sup J(u, v). (2.5)

v60 U

Since we established the existence of a maximizing u for any given v 6 0, it only remainsto estiblish the existence of a minimizing v to guarantee the existence of A.

2

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Let 0 be the adjoint variable associated with the maximization problem. For anyv 5 0, we need to select u to minimize

JoT{ r*'VI.X + X*VV 2u + 1u*IV3u + x'IV4v + 1v*1A'5v + ul'1 6v} dt (2.6)

From the maximum principle[10], which in this case is also a sufficient condition for opti-mality because of the uniqueness of the optimal u, the Hamiltonian is given by

1 .1 .1 .H = { + x*Wti + 1u*T U + X*WIv + 1 *vR V + u*11v+

2 2 20*{A(t)x + Bj(t)u + B 2(t)r}. (2.7)

The adjoint variable 0 satisfies

dO- = 11'x + ' u + I 4' - A*O (2.8)

x(O) = 0, O(T) = 0. (2.9)

Assuming that IV3 is invertible for all t E [0, T], the optimal controller is given by

= 1V-'(B*0 - IV*x - I'z,). (2.10)

LetA = A - BI1- 3 2iT,

BiI -'IB *

S 3

= v - U 11,l -1 2 , (2.11)

GI = B 2 - BI IV - 1TV6

G 2 = V 4 - IY 2 1'V 3 1176.

Thus we have

(x) (A b*) ( ) G (2.12)A o G

with

x(O) = 0, O(T) = 0. (2.13)

3

NADC-91077-6n

Let

(2.14)

M=(~~*)~(2.15)(0 -A*

N=(G) (2.16)N= G2'

U, = I4-'(-T 2 * B;) (2.17)

U,, = -W 3- 1 W 6 , (2.18)

an(lL =(IT 0). (2.19)

We define the matrices Q1, Q2, and Q3 by

Q, = UIV1 U, + U*U +- UWU* + U*WL+ UZ, (2.20)

Q2 = UI' 2U, Uzli.'v + ULI4 + U*1V 6. (2.21)

t'=, 3 w" - 2U, , W 6 . (2.22)

The system give:, by (2.12) can be written as

= 4(t)( + (t), (2.2j)

withx(O) = 0, O(T) = 0, (2.24)

and v needs to be selected to minimize the cost

fT !v*(t)R(t)v(t)dt(22

fJ{ *(t)QI(t)C(t) + (*Q2T, + V*Q3z} dt

We now investigate the conditions under which a minimizing v exists. Note that if

v = 0, the denominator of (2.25) is zero and that the minimum value of (2.25) over v 7 0is A.

THEOREM 2.1. Consider the system given by (2.23) and (2.24). Assume that there exists

a? E L 2 (0, T) for which 0 < fo { W Q (*Q 2 V + ,*Qv} dt <cc. Let

Tf i t,*R v d tA = inf . (2.26)

,,Vo ff { 2,*Q, + (Q2v + iv*Q3v} dt

4

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Also assume that R(t) - AQ3 (t) > a > 0 for all t E [0,T] and (Q Q2 is positive(Q2 0 )osemidefinite. Then there exists vo C L 2(0, T) which minimizes (2.25). Also, the niinimmnvalue of (2.25) is strictly positive.

Proof. Since (2.25) is invariant under scaling of v, it suffices to consider only those 1-

for which f0 T{ *Ql + 4*Q2v + t'*Q 3v} dt = 1. Let {z'} be a sequence in L 2(0. T) suchthat li,c T 1t,*R,. v, = A with T

tdt 2 A with f { (Qci,, + *Q2 v, + V,*Q 3 ,dt = 1 for each i.Here (i is the response of

= M(t)(, + N(t),,, (2.27)

withX,(O) = 0. 0,(T) = 0. (2.28)

Since {c,} is bounded in L 2 (0,T), a subsequence. still denoted by {Z,} convergesweakly to some v0 C L 2 (0,T). Also we can select the subsequence such that {(,(O)}.{f? t, Q3 v.dt } {f 0 Tv,,, dt} are convergent. Let ,(0) - . We have

t(i(t) = (0) + 1(t., T)N(r)rj(7) dr, (2.29)

where 4,(t, 7) is the transition matrix of (2.27). Let ( 0 (t) satisfy

to - 1Uk (,),o + N(t ), C O(a) ,. (2.30)

It is clear that by the weak convergence of {v,}, j(t) converges pointwise to '0 (t). From(2.29), it follows that for sufficiently large i. the responses (i(t) are uniformly bounded.By the Lebesgiie dominated convergence theorem, we conclude that

T( Tdo t)QIt)((t)dt , fo (,(t)Q,(t) o(t)dt. (2.31)

Also we have

iQ i dt d - ( - o)*Q2 r, dt + f o Q, i - vo) dt. (2.32)

It can be easily shown that the right side of (2.32) goes to zero as i 0C. Thus

hlr Q21', df j (,0Q 2r0 dt. (2.33)

NADC-91077-60

Note that+Ql 1 C¢ 2 + -v'Q 3 ,,} dt = 1Vi, (2.34)00 20 '1

jTT 1 * QT VT (f (oQlo+(oQ2o+ V Q3vojdt=1-lim -tiQ3 t'i dt+ -t'oQ 3 to dt. (2.33)

We already noted that limi- f T VK . dt exists. As a consequence of weak convergence

Tv0Q 3to dt <rnl j v*Q 3vi dt. (2.36)

We claim that the right side of (2.35) is strictly positive. Otherwise by the positive

semidefiniteness of (1 Q2) and by (2.34), (2.35) and (2.36), it follows that ro = 0 and( Q* 0lioi, f T ]

i 2 7vQ 3 t', dt = 1. From the assumption that R - AQ3 > a > 0, we have

1 1 l. o-viRci > A iQ3 1,'i + -v ,. (2.37)

-- 2

Integrating L r', ;ides from 0 to T and letting i go to cx, we get A > A + 3, where 3 > 0.This contradiction shows tAt

f*' 1 1

,j (Q 0 + C+Q2vO + tvO*Q 3v0} dt > 0. (2.3S)

We now show that

,T R,'o dt

S <. 2. 39)oI Ql(0 + o*Q2 O + VOQ3ZO} dt

Since lim,. fT 1 ,, Riv dt = A, we only need to show that

T T o Rvo dtlira t,*/zti dt - 7' 2 dt > 0. (2.40)

2 fo 2'{ 1 (* IQ 1 ( ;Q21', '*Q1',} -t

Indeed the numerator of

fT I dfo Qjro dtlim -t',Q1 z, dt - 0 (2.41)6 ,,*Q3,', t+ 7" 1'o Q3 1o (it

NADC-91077-60

, given by

nr nri (R - AQ3 ),', dt - -) - oQ3)zodl. (2.42)

Since R - AQ3 > 0, as a consequence of the weak convergence of {t',} to v, the abovequantity is nonnegative.

Also A needs to be greater than zero. Otherwise v0 = 0, which implies that thecorresponding unique optimal u = 0. This makes the response (0(t) = 0, and hence thedenominator of (2.25) is zero. contradicting (2.38). El

3. INFIMAL H., NORMIn the previous section, the minimax problem was converted into a minimization

problem. The resulting system was

=M()c + N(t), (3.1)

withx(O) = 0, O(T) = 0. (3.2)

where 7, needs to be selected to minimize the cost

f 7'1 ,(t)R (t)r,(t)dt.3 3

Jo { "'(t)Q(t) + (O(t) + Q(t)Q2(tW(t) + 1'*(t )Q3(t)(t)} dt

We now state the conditions that are satisfied by an optimal r(t).

TEoHEm 3.1. Consider the system given by (3.1)-(3.3). Assume that R-AQ, is invertiblefor allt C [0. T]. If r0 (t) minimizes (3.3). then there exists a nonzero p(t) (p*(t) q*(t) )*s!-ch that

=- .1)*p- AQI - AQ, (3.4)dt

where p(t ) and q(t ) are components of the adjoint vector corres)onding to x(t) and 0(t)respectively, such that

x(O) = 0, O(T)0, (3.5

p(T) = 0, q(0) = 0,

where

A = if T (3.6), , fo { Q Il + C*Q2,' + ,"Q3 } d

r,(t) = (? - AQ:)-'{AQ' + \-p}. (3.7)

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Proof. If u0 (t) minimizes (3.3), then it also minimizes

,]() 111= L c*Rvdt-A f 1(*Q + Q + 1v*Q3 r}dt. (3.8)

The theorem now follows from the mainmum principle[10]. 0Let

,Al = Al + AN(R - AQ3)-'Q>, (3.9)

= N(R - AQ3)-IN*, (3.10)and

£ -AQ1 - A 2Q 2(R - AQ3 1) . (3.11)

The variables satisfy a two point boundary value problem given by

(Q L)(I ). (3.12)

wvith

x(0) =0, O(T) 0. (

p(T) = 0, q(0) 0. (3.13)

We now give a criterion for the estimation of A. Notice that A = mineo max, J(u., r)

and gives a Ineasurc of performance of the optimal controller under worst-case conditionscorresponding to to(t). In the Ho case, the evaluation of A would entail the 5-iteration.

TiEOREM 3.2. Let A be the smallest positive value for which the boundary value problem

given by (3.12) and (3.13) has a solution ((,p) with fo Ql(+(*Q2t'+ Q3?ldt > 0,

where 1" = (R - AQ 3)-I {AQ*> + N*p}. Then A is the minimum value of (3.12), ((, p) is anoptimal pair and v ( R- AQ3)-{AQ*( + N t p} is the worst exogenous input.

Proof. It is clear from Theorem 3.1 that if vo(t) minimizes (3.3), then it satisfies (3.12)

and (3.13), with A being the minimum value of (3.3). Now suppose (C.p) satisfies (3.12)and (3.13) for some A. Let r = (R-AQ3 ) - {AQ,+N*p}. In the following equations ( , )denotes an inner product.

We have

jT ((R - AQ 3 )rv)d T (AQd (10 dt + IT(N* p,r) dt. (3.14)

By equation (3.1), the second integral of (3.14) can be written as

I' TT(A '*P, (It = ( .A _v ) d(it = 1 (p. -Al )Sdt. (3 1 )

NADC-91077-60

An integration by parts and equations (3.4) and (3.5) yieldTo ToT(P, - M)dt = A (Qc,)dt + Aj(¢,Q 2 v)dt. (3.16)

Substitut-ng (3.16) in (3.14), we get

v*R dt = A + *Ql(+2(*Q2v + vQ 3v} dt. (3.17)

Thus, the cost associated with v is A. Hence, if ((, p) is a nontrivial solution of the bound-ary value problem given by (3.12) and (3.13) for the smallest parameter A > 0 with

T1f{*Qcl + (*Q2 V+ ,*Q 3zv} dt > 0. then A is the optimal value and (v, p) is an optimal

pair. 0Note that the boundary value problem (3.12)-(3.13) has a solution with a nonvanishing

denominator for (2.4) for at most a countably infinite values of A. Theorem 3.2 gives asufficitit condition for an exogenous input to be optimal. Equations (3.12)-(3.13) andTheorem 3.2 completely characterize the worst exogenous input.

The criterion in Theorem 3.2 can be used to devise computational tools for the eval-uation of the infimal H, norm in the finite horizon case. Our computational experienceshows that for time-invariant problems, the infinal H_ norm in the finite horizon caseapproaches that in the infinite horizon case as the final time T becomes large.

4. COMPUTATION OF A

Making use of the transition matrix, the solution of (3.12) can be expressed as

0(t) = 2 1 (t, 0) 0 22 (t, 0) 0 2 3 (t, 0) 0 2 4 (t, 0) 0(0)4

p(t) 63 1(t, 0) 03 2 (t, 0) 6 3 3 (t, 0) 634 (t, 0) A (4.1)

q(t) ) \(P41(t, 0) 042 (t, 0) ( 4 3(t, 0) P44 (t, 0) q(O)

The boundary conditions given by (3.13) yield

03 2(T, 0) o33 (T, 0) 0(0)) -0. (4.2)

Let_ (22 0223 (4.3)

(032 033)

In view of (4.2) and (3.12)-(3.13), we have det (0'(T, 0)) = 0 if and only if the solution (,. p)of (3.12)-(3.13) is not identically zero. Thus, we need the least positive A which makes

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det( (T,0)) = 0 and the denominator of (2.4) positive. This can be obtained by doing asearch with A over an interval on which there is a change in the sign of the determinant.

We found the following algorithm to be numerically more stable since numbers oflesser magnitude are involved in the computation of the transition matrices in (4.4). Wehave

((T) (T )O(-,0)p(O) (4.4)

Let61' 12 3 4

: 2T 22= (4.5)€-1 (T, T 1 452 61 2 633 634

\41 42 43 64

and

-~ V 1/ 2 "13 /4

2T 0) = V32 V33 34 . (4.6)

V41 V42 V43 4 4

Making use of x(0) = q(0) = O(T) = p(T) = 0, we have

/ll 14) / "12 V132 2 (T) = v22 23 (0(0))61 64 q(T) - V32 V33 p(O)

41 44 \ V42 V43

The above equation has a nontrivial solution if and only if

7 'l ' 14 V12 V/13

det 21 24 V22 V23 (4.8)d 31 64 V32 133

( 41 44 V42 V43

Thus, we need the least positive A which makes the above determinant zero.

5. A DIFFERENTIAL EQUATION FOR AFot simplicity we derive a differential equation for A only in the case where W 4 =

WV5 = W 6 = 0. Note that this makes Q2 = Q3 = 0. Thus equations (3.12) and (3.13) canbe written as

= M( + NR-N*p (5.1)

= - M*p

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withx(O) = 0,O(T) = 0,

p(T) = 0, q(O) = 0. (5.2)

Now assume that the final time is changed to T + AT where AT is an elementalincrement. The solution of the above boundary value problem can be extended to [0, T +AT]. Suppose and P, are the elemental variations in ((, p) owing to the increment ATin T. That is, (( + (1,p + Pl) is the new optimal pair. Also denote the variation in A byA A. We have

6 = M i + NR-N*pl (5.3)=i = -AQI(I - AI*pi - AAQi(

withx1 (0) = 0,0 1(T + AT) = -O(T + AT), ()p (T + AT) = -p(T + AT),q(O) = 0.

THEOREM 5.1. As a function of T, A satisfies

dA -A(*(T)Ql(T)((T) - 2(*(T)M*(T)p(T) - p*(T)N(T)R-(T)N*(T)p(T)

dT f T (*Qdt (5.5)

Proof: From (5.3),

fT+AT T0 AT

(*p, dt = - {A(*QT l + (*MA*pl + AAC*QiC} dt. (5.6)

By an integration by parts,

f T+AT IT+ATS(* j dt = (*pl (T + AT) - 1 T+ (A l*pl + p*R-1N*pl} dt. (5.7)

From (5.6) and (5.7),

IT+AT fT+AT1 T AT(*Qil + AA *Q1 (} dt = -(,*pI(T + AT) + j p*NR-N*pl dt. (5.8)

From (5.1), the first integral on the left side of (5.8) can be written as

IT+AT fT+ATA(*Q,(, dt = - (TA ± *p)*( dt. (5.9)

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Integrating the right side of (5.9) by parts, we get

T+AT T+AT

A(*Qi(I dt = -p*(,(T + AT) + p*NR-N*pdt. (5.10)

Substituting (5.10) in (5.8), we get

AA f (*Q( dt = p*(T + AT)(,(T + AT) - *(T + AT)pi(T + AT). (5.11)

We have

p*(i(T + AT) - ,*p,(T + AT) = p*(T)(i(T + AT) - (*(T)pl(T + AT) + o(AT)

= -q*(T)O(T + AT) + x*(T)p(T + AT) + o(AT)

= -q*(T)d(T)AT + x*(T)3(T)AT + o(AT)

= -p(T),(T)AT + (*(T),(T)AT + o(AT) (5.12)

From (5.11) and (5.12). we get (5.5). 0

6,. EXAMPLES

The above theory is useful in the comiputation of the infimal H, norm. This problemis still being researched actively in the case of both static and dynamic controllers and thereare a variety of algorithms in the literature. In the examples given below, we compute theminimum H, norm given by (2.4) as the final time T varies. The programs were writtenusing PC-MATLAB and the least positive A which satisfies equation (4.8) was found. Theinfimal finite-time H, norm -y is 1/v/A.EXAMPLE 1. W~e consider the tracking example from [11]. In this case

A= ( 0 0 0 0

0 0 2 1 0

0.0081 -0.045 -0.0451' =(-0.045 0.25 0.25 , 0 ='1 1

-0.045 0.25 0.25 /

I4 = -0.05 , = 0.01, U" = 0, R=1.-0.05

The results are given below.

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TABLE 1: Results of Example 1T 1 __ =_I_/_ V_\5 17.6023 0.2384

10 16.3113 0.247615 15.9802 0.250220 15.7944 0.251625 15.7442 0.252030 15.7224 0.2522

EXAMPLE 2. This example, taken from [12] has

A 1 0 -2 - Bi 0 0 B2 = 1

_- - 1 1 0 1(0)

with1 0 1 0

-~, 1 0 1 0 0 1'0 0 0 0)

and zero entries in V2 , 1V4 , TV5, and 1V6. Table 2 gives the numerical results.

TABLE 2: Results of Example 2T A 1_ 1/_v_5 3.8975 0.506510 0.9220 1.041415 0.7001 1.195120 0.6722 1.219725 0.6681 1.2234

EXAMPLE 3. The last example is also taken from [12]. In this case

0 1 4 -4 1) (00)

A= 0 1 - 1 0 B= 0 0 B2 I

1 -1 -1 0B 021l 2 1 -2 -2 2

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withR~,1I 0 0 0 0),B

00 0 0 0 1 0

00 0 0 0

with the rest of the matrices having zero entries. The results axe given in Table 3.

TABLE 3: Results of Example 3

T A y = 1/vX5 0.89262204 1.05843988

10 0.87538735 1.0688084215 0.87477568 1.0691820318 0.87476217 1.06919029

7. SUBOPTIMAL PROBLEM FORMULATIONThe main contribution of the remaining portion of the report is the derivation of

the suboptimal controller in the time-varying case for a general performance index. Weconsider a generalized finite horizon suboptimal H, problem. An expression for a statefeedback controller is given in terms of solution of a dynamic Riccati equation. Also anexpression for a suboptimal output feedback controller is developed in terms of solutionsof two dynamic Riccati equations. Throughout the report an objective has been to deriveresults in as general a case as feasible. The formulae for the synthesis of the suboptimalH,, controller are summarized in Section 5 and these can be programmed easily on adigital computer to synthesize a suboptimal H controller. In the time invariant case, ifthe final time is sufficiently large, the solutions of the dynamic Riccati equations convergeto the solutions of the corresponding algebraic Riccati equations.

Let the n-dimensional time-varying system be given by

i = A(i)x + B,(t)u + B2 (t)v, x(to) 0, (7.1)

z = C(t)x + D(t)u + E(t)v, (7.2)

y = C2 (t)x + D 2 (t)u + E2 (t)v. (7.3)

Without loss of generality, let to = 0. Also let

A, = maxmin f4v*Rvdt)U fior T !z*IWz dt'

where R and W are assumed to be positive definite and the superscript * denotes a matrixor vector transpose. Computational techniques for the evaluation of Aopt are given inSection 4. The problems addressed in the report can be stated as follows.

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Problem 1. Given A < Aopj, find a full state feedback controller, if it exists, for which

rin fT v*Rv dtmin f -> A.0 T*1zdt

Problem 2. Given A < Aopt, find an output feedback controller, if it exists, for which

min fT *Rv dtmin fo> A.Vo$0 fT z. z dt

8. FULL STATE FEEDBACK PROBLEM

Consider the performance criterion

oT v * Rv dt - A z *IVz dt. (8.1)

We will first find a saddle point (u0 ,t v) with respect to the criterion (8.1). The motivationfor finding the saddle point is so that we can construct a suboptimal H, controller on thefinite interval [0, T].

The functional (8.1) can be written as

J(u. v) = _ I *Rvdt - A 1-X*HVIx + X*1'2u + 1_u*l1-U1

+X*11 4 v + V*WV + u *W6v} dt. (8.2)

Given u(t), let v° maximize (8.2). The following lemma characterizes v0 (t).

LEMMA 8.1. Let A be such that R - AIW5 is positive definite for all t E [0, TI. For a givenu, if v°(t) minimizes (8.2), then tl.ere exists a nonzero 71(t) such that

d r- A*r - AVlx - AgW2 u - AW 4 v ° , 77(T) - 0, (8.3)dt

andv 0 (t) = (R - AT 5 )-' {Bt + A/7x + AV6*u.} (8.4)

Proof. By the maximum principle[10], there exists an adjoint response 17(t) such thatthe Hamiltonian

1 .1 .- ,1 WH = A{-1x*Wix + x*W2u + Iu *14'U + X*T' 4V + -vW1v* V + u* 6 t'}

2 2 2-1 v *Rv + 77* {A(t)x + B (t)u + B 2 (t)v. } (8.5)

2

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is maximized almob everywhere on [0, T]. Satisfaction of H= 0 yields (8.4). The adjointvariable q satisfies

- OH -A' - A x - "I,"u - AW 4 v0 . (8.6)dt Ox

By the transversality condition, 77(T) = 0. [1In a similar manner, we can get an expression for an optimal u0 (t) which maximizes

(8.2) for given v and A.

LEMMA 8.2. Consider the system given by (7.1). Assume that WV3 is positive definite forall t E [0, T1. For a given v, if u0 (t) maximizes (8.2), then there exists a nonzero V,(t) suchthat

- A + W x + W 2u 0 + VVv, 0(T)=0, (8.7)dt

anduO(t) = TV17' {B*/- WV*x - 17v}. (8.8)

Proof. Proof is similar to that of Lemma 8.1. DSimultaneous solution of (8.3), (8.4), (8.7), and (8.8) yields a saddle point solution

(u0 , v0 ) for the functional given by (8.2). We now express the above minimax solution ina simpler form.

From (8.3) and (8.7), at a saddle point solution (u°, v°), we get

d(Ai, + ri) = -A*(t)(AV, + 77), AV(T) +rj(T) = 0. (8.9)dt

It follows thatAi/(t) + q(t) = 0, t C [0, T]. (8.10)

Thus the saddle point solution is characterized by

A*V + H" x + Tu ° + T,°, (8.11)dt

u(t) = 91-' {B* - 7W*x - W 1;}, (8.12)

v°(t) = A(R - AW)- 1{-B*', + W*x + W6*uO}. (8.13)

We define the full state feedback controller the following way. Assuming that theinverse of WV3 + AW 6(R - AW)-W * exists, let

Q (R- AW)-', (8.14)A = {W 3 + AW6f2W}-', (8.13)

U1 = A(B; + AlW6frB*), (8.16)U2 = -A(W* + ATV 6 ' 4*), (8.17)

Q= Ai(-B* + W*U,),,(.1,,

V2 = AQ(W* + W6U 2 ). (8.19)

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Substituting (8.13) in (8.12), we get

U0 = UI-'+ U 2x. (8.20)

Substituting (8.20) in (8.13), we get

V0 = + V2x. (8.21)

Let f be arbitrary. On [f, T], let

=(t) P(t)x(t). (8.22)

On [c, T], we get

P + P(A + BIU 2 + B 2 1 2 ) + (A* - W2U1 - I' 4 V)P

+P(B, U1 + B 2V 1 )P - (W 1 + W 2 U2 + W4 1) = 0, P(T) = 0. (8.23)

Define the feedback controller and the exogenous input by

uo = U, PX + U2 x, (8.24)

vo = VI"PX + V2x. (8.25)

We now show that the performance of this feedback controller is greater than A.

THEOREM 8.1. Consider equations (8.23) and (8.24). Then for this controller

Tn if v* Rv dt

inf fT 2 *.v d > A. (8.26)v -O fT ZZ dt

Proof. An elementary calculation shows that in equation (8.23), B 1 U1 + B2 V, andTVl + IV2 U 2 + 114 1' are symmetric and

(A + BU 2 + B 2 V2 )* = A* - 9' 2 U, - 14741,. (8.27)

Thus P is symmetric. Since (8.20) and (8.21) follow from (8.12) and (8.13), we have

Uo = 7- { Bj* Px - W2* x - W6 vo}, (8.28)

Vo = AQf{-B2Px + W4*x + W6uo}. (8.29)

Since P(T) = x(O) = 0, we have

[T d (r Px)dt = 0 (8.30)

J0 dt'

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Sincei = Ax + Bluo + B 2V = (A + B 1 UIP + B 1 U2 )x + B2t,, (8.31)

where v is an arbitrary function of time, we get after some algebra

d(x*Px) = x*Px + 2x*Pi

= * + x*W 2 uo + x*Wlvo - x*PB2 vo + x*PBiuo + 2x*PB2v.(8.32)

From (8.28) and (8.29), we get

B*Px = Wuo + Hr*x + 11'6 v0 , (8.33)Q' -1 t',0

B*Px = I-+ *x W 0, (8.34)

Substituting (8.33) and (8.34) in (8.32), we get

dd(x*Px) = x*ltix + 2x*IV1uo + u*)TVuo + 2x*TV4v

r*)*-'(v 0 )+ 2ulV' t + ( (8.35)

From (8.35) and (8.30), we get

JT v*Rv dt - A j z*'Wz dt = (v - vo)*Q-l(v - vo)dt. (8.36)

Note that in the above equation v is an open loop function of t, whereas v0 is a feedbackfunction of x. Note also that Q-' is positive definite on [0, T]. Since the map v - v - v

and its inverse are bounded, it follows that there exists 6 > 0 such that

(v - vo)*Q (v - vo)dt > 6j v*Rvdt. (8.37)

From (8.36) and (8.37), we have (8.26). D

9. OUTPUT FEEDBACK CONTROLLER

Consider again the system given by

x = A(t)x + B,(t)u + B 2 (t)v, (9.1)

z = C(t)x + D(t)u + E(t)v, (9.2)

y = C 2(t)x + D 2(t)u + E 2(t)v. (9.3)

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Let= C2 + E2VIP + E21"2, (9.4)

where V, and V2 are defined by (8.18) and (8.19). Assume that the controller is of theform

4 = Aq + BI(U1 Pq + U2q) + B 2 (1'jPq + V2q) + L(Cq + D2u - y), (9.5)

u = U1Pq + U2q, (9.6)

where the observer gain L needs to be determined.Let

Uo = U1 Px + U2x, (9.7)

v0 = TPx + V2x, (9.8)

r = u -u 0 , (9.9)

w = v - v0, (9.10)

e = x - q. 9.11)

Let the control be given by (9.6) and let P be the solution of (8.23). We have

x z Ax + BI(U P + U2 )q + B 2 , (9.12)

where v is an arbitrary function of time.

LEMMA 9.1. For the above system, we have

v*Rrdt - =z*Wzdt = *Q-1udt

-AI r*T 3 r dt - 2A r*16uwdt. (9.13)

Proof. We have

x*PX +2x*P' = x* IW +x*IV2 uo +x W*IVo +x*PB(2U-Uo)+ x*PB 2(2Vt,-Vo). (9.14)

From (8.33) and (8.34),

B*Px = V3uo + 14*x + Wvo, (9.15)

BPx = A + 117* x + TI'V6. (9.16)

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incorporating (9.15) and (9.16) in (9.14) and rearranging, the right side of (9.14) can bewritten as

x*lllx + 2x*12u + u*1lau + 2x*'114 V + 2u'6i,,A

(v - ,Z)f -'(' - ')-(U -uo)*1V3(u - uo)+ ( - 2(u - Uo)6( v - o). (9.17)

Since

T +(xPx)dt = j {x'Px + 2x*Pil dt =, (9.1S)

the result follows. El\e want to ultimately show that the right side of (9.13) is larger than < .7 i*Ridt

for some 6 > 0. It easily follows that ( = x - q ani r = u - uo satisfy

= (A + LC')( + (B 2 + LE)w, (9.19)

r =Th, (9.20)

where

.4 + B 2 1"P + B2 V21 (3.21)

= -(U 1P + U2). 9.22y.

Note that C is defined by (9.4) and L is the gain of the observer. The right side of (9.13)can be written as

f u*Ru'dt- Ao j Wizdt. (9.23)

where (see (7.2))z1 = Dr + Eu, = DB + Ew. (9.24)

XVe determine L by considering the dual of (9.19) and (9.24). Our ultimate goal is to showthat for the L to be iioson, the controller given by (9.5) and (9.6) is suboptimal.

Let r = -t. The dual system can be defined on [-T,O] as

=7 + C*fi + B*D*t,, (9.25a)drz.. j B 2- + E *fi + E'ti,, (9.25b)

and the functional corresponding to (9.23) is written as

J" 1'zi1U- 'adr- AJ T R-'- dr. (9.26)

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Let us write (9.26) as

1 ' I - , d T - A lu ±+

+*11'4,1, + ±, IV,5,7 + f, * W6,t } dr. (9.27)

We now find a saddl- point solution for the functional in (9.27). The saddle point solutionwill be useful in defining the gain L of the observer.

This problem is completely analogous to the full state feedback problem of Section 3and we can write the solution by inspection. Equations (9.25a) and (9.27) are analogousto equations (7.1) and (R.2) respectively. Assume that TV3 and W -1 - AIV5 are positivedefinite and .3 is the adjoint varial)le. Observing (8.11 )-(8.13), the saddle point solution ischaracterized by

-.4 + Ii' + Ii2 fi + Ii' 3(0) 0, (9.28)dri- C3 - * 6,}. (9.29)

- A(1--' - Al1-'{-D3 + Il'T" + i" }. (9.30)

Assume that Ii3 + Ali;( -11 - Ali's) 11* is invertible. The equation a anlogous to(S.14)-(S.19) are

= 1 " - AlV;) - (9.31)

F {113 + AIV441t}-'. (9.32)

, r(C' + AWAI;DB). (9.33)

S 2 = -F(Wi-" + A~iT64i)Ti4 . (9.34)T, Aq,(-DB + I''S,). (9.35)

T2 A--(IV* + 116" 2 ). (9.36)

Sub~stituting (9.30) in (9.29). we can write 5i and 6- as

f, = S, 3 + S2 , (9.37)

6, = T, 3 + T2i. (9.38)

Letting

3 Y C (9.39)

on [-TO], we get tie Riccati equation

dY"- (A - IV2S, - IV4T, )Y + Y(A* + C"S2 + B*D*T2 )

dr

+±(C0 1 + B*D*Ti )Y - (Ii, + li' 2 S 2 + I3T 2 ), )'(0) 0, (9.40)

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which is analogous to (8.23). It can be easily verified that (9.40) is symmetric.Define tio and tio' by

Uiu = S,1 " + S2j, (9.41)

Uo = T 1Y7+ T2&. (9.42)

LEMMA 9.2. Consider equations (9.40) and (9.41) and assume that at f = , given c > 0there exists T > 0 such that for any ti, 5 0 and all T > T, j (-T)Y(-T)((-T)_

Cf2T itIY'-1 dr. Then there existq 6 > 0 such that for all T> T,

f 0 0oTj'*1 -jd7 -A fJ R-X51 d- > 6 fT *I'-'t d. (9.43)

Proof. Following similar reasoning as in the proof of Theorem 8.1. we can write theequation analogous to (8.35) as

+C(* z + 2 T172,, + iTV 3i 0 + 2zP117, ' - -51z,

-. -.- f*iv-1lti+ (1i - tilo)*D (7 - ZP0 (9.44)+2i01V;Z -u A + A (.4

Integrating both sides from -T to 0., we get

J7'*I V-16 dr - A z*R 'zl d7-

T (6,- o)* (z - o)di + *(-T)(-T)(-T). (9.45)

Let c > 0 be sufficiently small. Since the map ti --+ zi'- t and its inverse are bounded,there exists 6 > 0 such that

(f' - ii'0)*D-1(z, - tN'o) d -, (V + AE) j w dr-. (9.46)

Now by the assumption on *(-T)Y(-T) (-r), equation (9.43) follows, if T > T UWe now go back to the original problem and reverse time in (9.40).

T1EOR0EM 9.1. On [0, T]. let

(A - i'2S1 - li'4 Tl )IY + Y(A* + * S2 + B* D*T2 )

+Y(C*Sl + b*D*T )Y" - (WItl + 11'2 S2 + li4T 2 ), Y(0) = 0. (9.47)

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L = (S 1Y + S2 )*. (9.48)

Consider (9.12). Let t be as defined in Lemma 9.2. If T > T, there exists 6 > 0 such that

Tv*Rv dt - A IT z*Wz dt > T v*Rv dr. (9.49)

Proof. From equation (9.43) of the dual system, we deduce that if T> T

w*Rwdt - A z14rzl dt > b, w*Rw dt, (9.50)

for some 61 > 0. Since the map w --+ v is bounded, there exists 6 > 0 such that

j w*Rwdt - AJ 4iVz, dt > v*Rvdt. (9.51)

From (9.13), (9.23), (9.24), and (9.51), we get (9.49). ElThe above theorem shows that for the controller defined by (9.5) and (9.6). the per-

formance is greater than A. In fact from equation (9.49), we have

inf f f R dt

inf > A. (9.52),7o fT z*5 z (it

10. SUMMARY OF RESULTSThe system is given by

j = A(t)x + B 1 (t)u + B 2 (t)v, x(O) = 0, (10.1)

z = C(t)x + D(t)u + E(t)v, (10.2)

y = C 2(t)x + D 2(t)u + E 2(t)v. (10.3)

Let

Aopt = maxmin f °T v*Rvdt (10.4)U V:o foT z*Wzdt

and let A < Aopt. Also, let W 1 ,.' ,W 6 be defined by

z*Wz = x*TV x + 2x*1V2u + u*11 3 u + 2x*W4t + v*I'5v + 2u116 r. (10.5)

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The relevant controller equations are

Q = (R - A )-', (10.6)A = {W 3 + ,d4 6 QV1*}, (10.7)

U1 = A(BI* + Af117 6 B*), (10.8)

U2 = -A(W7* + AllQW4*), (10.9)

V = Af(-B2 + 11'U 1 ), (10.10)

V2 = AM(TV4" + 1 '6 ) U), (10.11)

+ P(A + B, U2 + B 2V2) + (A* - IV2 U - I' )P

+P(BU, + B 2 I 1 )P - (ITW + W 2U2 + W4 1') = 0, P(T) 0. (10.12)

Note that the above equation is symmetric.Let

= B* + E2i + E*z., (10.13)

and let U-1," ', li 6 be defined byz,= *117,{ + 2E*-V iii +± 2EauV+2 ±xt,+ '*Ii, + 2i*Ii- z. (10.14)

The relevant observer equations are

) = (1- 1 - A,,*5)- 1 , (10.15)

F = {143 + AW64)1 6 -1 , (10.16)

A = A + B 2VP + B 2 12 , (10.17)

= -(U 1 P + U2 ), (10.18)

= C2 + E 2 VIP + E 21'", (10.19)

S1 = F(C -+ A4)D!b), (10.20)

S2 = -r(W"2 + AWO).i '), (10.21)

T, = A-t(-Db + WgV*S1 ), (10.22)

T2 = A )( + W6 S2). (10.23)

1" = (A - lab2S 1 - 4T,)Y + Y(4A* + C'S 2 + B'D'T2 )

"-'(C'S1 + b*D*Ti)Y - (WI - li2S2 + W17 4 T 2 ), F(0) = 0. (10.24)

Note that the above equation is symmetric.

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The suboptimal controller is given by

4 = Aq + BI(U 1 Pq + U2q) + B 2 (ViPq + V2q) + L(Cq + D2u - y), (10.25)

L = (S 1 Y + S2 )*, (10.26)

u = (UiP + U)q. (10.27)

11. CONCLUSIONSIn this report we treated the suboptimal finite horizon H, control problem. A differ-

ential equation for the measure of performance is derived. The report also illustrates theusefulness of the finite horizon techniques in computing the infimal H, norm in the infi-nite horizon case. An expression for a suboptimal finite horizon H, controller is derivedin a generalized case. The general case has been treated directly without the utilizationof transformations. The ouput feedback controller is synthesized via a controller compo-nent and an observer component. A summary of all the design equations is given andthese equations are easy to program on a digital computer. In the time-invariant case thedynamic Riccati equations involved in the design usually converge to constant matrices.The theory developed in this report is useful for the worst case design of flight controlsystems of aircraft in the presence of disturbances, commands, and sensor noise. The con-troller equations are in a form that is convenient for the design of flight control systemsof advanced aircraft, since the results are applicable for a very general case. Preliminaryapplications to flight control design have shown considerable promise and the next phaseof research will be directed towards the development of a systematic methodology for thedesign of a flight control system for an advanced aircraft. Further research also needs tobe done to assess the relationship of the weighting matrices to satisfactory performance.

12. FUTURE WORKSoftware is already in place to predict the best aircraft performance under worst case

conditions. The success of this portion of the software is demonstrated in Section 6 ofthis report by way of examples. Future work based on the theoretical results of the reportwould entail the following:

1. Making use of the theory developed in Sections 7-9, software will be developed tosynthesize both state and output feedback controllers.

2. Making use of an advanced aircraft model during landing, flight control laws willbe developed to achieve satisfactory performance in the presence of disturbances,commands, and variations of the model.

3. The performance of the aircraft will be compared with that achieved through othertechniques and the benefits, if any, will be demonstrated.

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REFERENCES

[11 J. C. DOYLE, K. GLOVER, P. P. KHARGONEKAR, AND B. A. FRANCIS, State-spacesolutions to standard H2 and Hoo control problems, IEEE Transactions on AutomaticControl, Vol. 34, 1989, pp. 831-8417.

[21 M. G. SAFONOV, D. J. N. LIMEBEER, AND R. Y. CHIANG, Simplifying the H the-ory via loop-shifting, matrix-pencil and descriptor concepts, International Journal ofControl, Vol. 50, No. 6, 1989, pp. 2467-2488.

[3] R. RAVI, K. M. NAGPAL, AND P. P. KHARGONEKAR, H1 control of linear time-varying systems: A state space approach, SIAM Journal on Control and Optimization,to appear.

[4] N. B. SUBRAHMANYAM, Necessary conditions for minimum in problems with non-standard cost functionals, J. Math. Anal. Appl., Vol. 60, 1977, pp. 601-616.

[5] M. B. SUBRAHMANYAM, On applications of control theory to integral inequalities, J.Math. Anal. Appl., Vol. 77,1980, pp. 47-59.

[6] M. B. SUBRAHMANYAM, On applications of control theory to integral inequalities: II,SIAM J. Control Optim., Vol. 19,1981, pp. 479-489.

[7] M. B. SUBRAHMANYAM, A control problem with application to integral inequalities, J.Math Anal. Appl., Vol. 81,1981, pp. 346-355.

[8] M. B. SUBRAItMANYAM, An extremal problem for convolution inequalities, J. Math.Anal. Appl., Vol. 87,1982, pp. 509-516.

[9] M. B. SUBRAHMANYAM, On integral inequalities associated with a linear operatorequation, Proc. Amer. Math. Soc., Vol. 92,1984, pp. 342-346.

[10] E. B. LEE AND L. MARKUS, Foundations of Optimal Control Theory, John Wiley,New york, 1967.

[11] B. A. FRANCIS AND J. C. DOYLE, Linear Control Theory with an H, OptimalityCriterion, SIAM J. Control and Optimization, Vol. 25, No. 4, July 1987, pp. 815-844.

[12] W. R. PERKINS AND J. V. MEDANIC, Systematic Low Order Controller Design for Dis-turbance Rejection and Plant Uncertainties, Report #WRDC-TR-90-3036, Wright-Patterson AFB, July 1990.

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